Hey, hot question real quick. I am doing Functions in Algebra and I kinda need help. So the function I have is **g(x)=x^2-9** and I have to evaluate **g^-1(x)**, but my question is how does the exponent go in front of the numbers? What does that mean? I know what to do with negative exponents but what do I do when they are in front of the numbers? How would I even write this?
g^-1 means "g inverse"
Clearly you have not studied the material.
For example, if
f(x) = 2x+5
g(x) = f^-1(x) = (x-5)/2
since
f(g(x)) = g(f(x)) = x
Great question! In this context, when the exponent is in front of the number, it indicates a function known as the "inverse function" or "inverse of the function".
To find the inverse function, you need to follow a few steps:
Step 1: Start with the given function **g(x)**, which in this case is **g(x) = x^2 - 9**.
Step 2: Replace **g(x)** with the variable **y** to represent the inverse function: **y = g(x)**.
Step 3: Swap the positions of **x** and **y**: **x = g(y)**.
Step 4: Solve the equation **x = g(y)** for **y** to get the inverse function **g^-1(x)**.
Now, let's apply these steps to your example, **g(x) = x^2 - 9**.
Step 1: Start with **g(x) = x^2 - 9**.
Step 2: Replace **g(x)** with the variable **y**: **y = x^2 - 9**.
Step 3: Swap the positions of **x** and **y**: **x = y^2 - 9**.
Step 4: Solve the equation **x = y^2 - 9** for **y** to get the inverse function **g^-1(x)**.
To solve for **y**, we need to rearrange the equation. Let's start by isolating **y^2**:
**x = y^2 - 9**
**x + 9 = y^2**
Then, take the square root of both sides:
**sqrt(x + 9) = y**
So the inverse function **g^-1(x)** is given by:
**g^-1(x) = sqrt(x + 9)**.
And that's how you find the inverse function when the exponent is in front of the numbers!